Theorem 163prm 13259

Description: 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

Assertion

Ref	Expression
163prm	⊢ ;;163 ∈ ℙ

Proof of Theorem 163prm

Step	Hyp	Ref	Expression
1		1nn0 9583	. . . 4 ⊢ 1 ∈ ℕ0
2		6nn0 9588	. . . 4 ⊢ 6 ∈ ℕ0
3	1, 2	deccl 9795	. . 3 ⊢ ;16 ∈ ℕ0
4		3nn 9471	. . 3 ⊢ 3 ∈ ℕ
5	3, 4	decnncl 9804	. 2 ⊢ ;;163 ∈ ℕ
6		8nn0 9590	. . 3 ⊢ 8 ∈ ℕ0
7		4nn0 9586	. . 3 ⊢ 4 ∈ ℕ0
8		3nn0 9585	. . 3 ⊢ 3 ∈ ℕ0
9		1lt8 9505	. . 3 ⊢ 1 < 8
10		6lt10 9919	. . 3 ⊢ 6 < ;10
11		3lt10 9922	. . 3 ⊢ 3 < ;10
12	1, 6, 2, 7, 8, 1, 9, 10, 11	3decltc 9818	. 2 ⊢ ;;163 < ;;841
13		6nn 9474	. . . 4 ⊢ 6 ∈ ℕ
14	1, 13	decnncl 9804	. . 3 ⊢ ;16 ∈ ℕ
15		1lt10 9924	. . 3 ⊢ 1 < ;10
16	14, 8, 1, 15	declti 9823	. 2 ⊢ 1 < ;;163
17		2cn 9377	. . . 4 ⊢ 2 ∈ ℂ
18	17	mullidi 8329	. . 3 ⊢ (1 · 2) = 2
19		df-3 9366	. . 3 ⊢ 3 = (2 + 1)
20	3, 1, 18, 19	dec2dvds 13210	. 2 ⊢ ¬ 2 ∥ ;;163
21		5nn0 9587	. . . 4 ⊢ 5 ∈ ℕ0
22	21, 7	deccl 9795	. . 3 ⊢ ;54 ∈ ℕ0
23		1nn 9317	. . 3 ⊢ 1 ∈ ℕ
24		0nn0 9582	. . . 4 ⊢ 0 ∈ ℕ0
25		eqid 2238	. . . 4 ⊢ ;54 = ;54
26	1	dec0h 9807	. . . 4 ⊢ 1 = ;01
27		ax-1cn 8272	. . . . . . 7 ⊢ 1 ∈ ℂ
28	27	addlidi 8470	. . . . . 6 ⊢ (0 + 1) = 1
29	28	oveq2i 6096	. . . . 5 ⊢ ((3 · 5) + (0 + 1)) = ((3 · 5) + 1)
30		5p1e6 9444	. . . . . 6 ⊢ (5 + 1) = 6
31		5cn 9386	. . . . . . 7 ⊢ 5 ∈ ℂ
32		3cn 9381	. . . . . . 7 ⊢ 3 ∈ ℂ
33		5t3e15 9886	. . . . . . 7 ⊢ (5 · 3) = ;15
34	31, 32, 33	mulcomli 8333	. . . . . 6 ⊢ (3 · 5) = ;15
35	1, 21, 30, 34	decsuc 9816	. . . . 5 ⊢ ((3 · 5) + 1) = ;16
36	29, 35	eqtri 2259	. . . 4 ⊢ ((3 · 5) + (0 + 1)) = ;16
37		2nn0 9584	. . . . 5 ⊢ 2 ∈ ℕ0
38		2p1e3 9440	. . . . 5 ⊢ (2 + 1) = 3
39		4cn 9384	. . . . . 6 ⊢ 4 ∈ ℂ
40		4t3e12 9883	. . . . . 6 ⊢ (4 · 3) = ;12
41	39, 32, 40	mulcomli 8333	. . . . 5 ⊢ (3 · 4) = ;12
42	1, 37, 38, 41	decsuc 9816	. . . 4 ⊢ ((3 · 4) + 1) = ;13
43	21, 7, 24, 1, 25, 26, 8, 8, 1, 36, 42	decma2c 9838	. . 3 ⊢ ((3 · ;54) + 1) = ;;163
44		1lt3 9480	. . 3 ⊢ 1 < 3
45	4, 22, 23, 43, 44	ndvdsi 12716	. 2 ⊢ ¬ 3 ∥ ;;163
46		3lt5 9485	. . 3 ⊢ 3 < 5
47	3, 4, 46	dec5dvds 13211	. 2 ⊢ ¬ 5 ∥ ;;163
48		7nn 9475	. . 3 ⊢ 7 ∈ ℕ
49	37, 8	deccl 9795	. . 3 ⊢ ;23 ∈ ℕ0
50		2nn 9470	. . 3 ⊢ 2 ∈ ℕ
51		eqid 2238	. . . 4 ⊢ ;23 = ;23
52	37	dec0h 9807	. . . 4 ⊢ 2 = ;02
53		7nn0 9589	. . . 4 ⊢ 7 ∈ ℕ0
54	17	addlidi 8470	. . . . . 6 ⊢ (0 + 2) = 2
55	54	oveq2i 6096	. . . . 5 ⊢ ((7 · 2) + (0 + 2)) = ((7 · 2) + 2)
56		7t2e14 9894	. . . . . 6 ⊢ (7 · 2) = ;14
57		4p2e6 9450	. . . . . 6 ⊢ (4 + 2) = 6
58	1, 7, 37, 56, 57	decaddi 9845	. . . . 5 ⊢ ((7 · 2) + 2) = ;16
59	55, 58	eqtri 2259	. . . 4 ⊢ ((7 · 2) + (0 + 2)) = ;16
60		7t3e21 9895	. . . . 5 ⊢ (7 · 3) = ;21
61		1p2e3 9441	. . . . 5 ⊢ (1 + 2) = 3
62	37, 1, 37, 60, 61	decaddi 9845	. . . 4 ⊢ ((7 · 3) + 2) = ;23
63	37, 8, 24, 37, 51, 52, 53, 8, 37, 59, 62	decma2c 9838	. . 3 ⊢ ((7 · ;23) + 2) = ;;163
64		2lt7 9497	. . 3 ⊢ 2 < 7
65	48, 49, 50, 63, 64	ndvdsi 12716	. 2 ⊢ ¬ 7 ∥ ;;163
66	1, 23	decnncl 9804	. . 3 ⊢ ;11 ∈ ℕ
67	1, 7	deccl 9795	. . 3 ⊢ ;14 ∈ ℕ0
68		9nn 9477	. . 3 ⊢ 9 ∈ ℕ
69		9nn0 9591	. . . 4 ⊢ 9 ∈ ℕ0
70		eqid 2238	. . . 4 ⊢ ;14 = ;14
71	69	dec0h 9807	. . . 4 ⊢ 9 = ;09
72	1, 1	deccl 9795	. . . 4 ⊢ ;11 ∈ ℕ0
73	31	addlidi 8470	. . . . . 6 ⊢ (0 + 5) = 5
74	73	oveq2i 6096	. . . . 5 ⊢ ((;11 · 1) + (0 + 5)) = ((;11 · 1) + 5)
75	66	nncni 9316	. . . . . . 7 ⊢ ;11 ∈ ℂ
76	75	mulridi 8328	. . . . . 6 ⊢ (;11 · 1) = ;11
77	31, 27, 30	addcomli 8472	. . . . . 6 ⊢ (1 + 5) = 6
78	1, 1, 21, 76, 77	decaddi 9845	. . . . 5 ⊢ ((;11 · 1) + 5) = ;16
79	74, 78	eqtri 2259	. . . 4 ⊢ ((;11 · 1) + (0 + 5)) = ;16
80		eqid 2238	. . . . 5 ⊢ ;11 = ;11
81	39	mullidi 8329	. . . . . . 7 ⊢ (1 · 4) = 4
82	81, 28	oveq12i 6097	. . . . . 6 ⊢ ((1 · 4) + (0 + 1)) = (4 + 1)
83		4p1e5 9443	. . . . . 6 ⊢ (4 + 1) = 5
84	82, 83	eqtri 2259	. . . . 5 ⊢ ((1 · 4) + (0 + 1)) = 5
85	81	oveq1i 6095	. . . . . 6 ⊢ ((1 · 4) + 9) = (4 + 9)
86		9cn 9394	. . . . . . 7 ⊢ 9 ∈ ℂ
87		9p4e13 9874	. . . . . . 7 ⊢ (9 + 4) = ;13
88	86, 39, 87	addcomli 8472	. . . . . 6 ⊢ (4 + 9) = ;13
89	85, 88	eqtri 2259	. . . . 5 ⊢ ((1 · 4) + 9) = ;13
90	1, 1, 24, 69, 80, 71, 7, 8, 1, 84, 89	decmac 9837	. . . 4 ⊢ ((;11 · 4) + 9) = ;53
91	1, 7, 24, 69, 70, 71, 72, 8, 21, 79, 90	decma2c 9838	. . 3 ⊢ ((;11 · ;14) + 9) = ;;163
92		9lt10 9916	. . . 4 ⊢ 9 < ;10
93	23, 1, 69, 92	declti 9823	. . 3 ⊢ 9 < ;11
94	66, 67, 68, 91, 93	ndvdsi 12716	. 2 ⊢ ¬ ;11 ∥ ;;163
95	1, 4	decnncl 9804	. . 3 ⊢ ;13 ∈ ℕ
96	1, 37	deccl 9795	. . 3 ⊢ ;12 ∈ ℕ0
97		eqid 2238	. . . 4 ⊢ ;12 = ;12
98	53	dec0h 9807	. . . 4 ⊢ 7 = ;07
99	1, 8	deccl 9795	. . . 4 ⊢ ;13 ∈ ℕ0
100		eqid 2238	. . . . 5 ⊢ ;13 = ;13
101	32	addlidi 8470	. . . . . 6 ⊢ (0 + 3) = 3
102	8	dec0h 9807	. . . . . 6 ⊢ 3 = ;03
103	101, 102	eqtri 2259	. . . . 5 ⊢ (0 + 3) = ;03
104	27	mulridi 8328	. . . . . . 7 ⊢ (1 · 1) = 1
105		00id 8468	. . . . . . 7 ⊢ (0 + 0) = 0
106	104, 105	oveq12i 6097	. . . . . 6 ⊢ ((1 · 1) + (0 + 0)) = (1 + 0)
107	27	addridi 8469	. . . . . 6 ⊢ (1 + 0) = 1
108	106, 107	eqtri 2259	. . . . 5 ⊢ ((1 · 1) + (0 + 0)) = 1
109	32	mulridi 8328	. . . . . . . 8 ⊢ (3 · 1) = 3
110	109	oveq1i 6095	. . . . . . 7 ⊢ ((3 · 1) + 3) = (3 + 3)
111		3p3e6 9449	. . . . . . 7 ⊢ (3 + 3) = 6
112	110, 111	eqtri 2259	. . . . . 6 ⊢ ((3 · 1) + 3) = 6
113	2	dec0h 9807	. . . . . 6 ⊢ 6 = ;06
114	112, 113	eqtri 2259	. . . . 5 ⊢ ((3 · 1) + 3) = ;06
115	1, 8, 24, 8, 100, 103, 1, 2, 24, 108, 114	decmac 9837	. . . 4 ⊢ ((;13 · 1) + (0 + 3)) = ;16
116	18, 28	oveq12i 6097	. . . . . 6 ⊢ ((1 · 2) + (0 + 1)) = (2 + 1)
117	116, 38	eqtri 2259	. . . . 5 ⊢ ((1 · 2) + (0 + 1)) = 3
118		3t2e6 9463	. . . . . . 7 ⊢ (3 · 2) = 6
119	118	oveq1i 6095	. . . . . 6 ⊢ ((3 · 2) + 7) = (6 + 7)
120		7cn 9390	. . . . . . 7 ⊢ 7 ∈ ℂ
121		6cn 9388	. . . . . . 7 ⊢ 6 ∈ ℂ
122		7p6e13 9863	. . . . . . 7 ⊢ (7 + 6) = ;13
123	120, 121, 122	addcomli 8472	. . . . . 6 ⊢ (6 + 7) = ;13
124	119, 123	eqtri 2259	. . . . 5 ⊢ ((3 · 2) + 7) = ;13
125	1, 8, 24, 53, 100, 98, 37, 8, 1, 117, 124	decmac 9837	. . . 4 ⊢ ((;13 · 2) + 7) = ;33
126	1, 37, 24, 53, 97, 98, 99, 8, 8, 115, 125	decma2c 9838	. . 3 ⊢ ((;13 · ;12) + 7) = ;;163
127		7lt10 9918	. . . 4 ⊢ 7 < ;10
128	23, 8, 53, 127	declti 9823	. . 3 ⊢ 7 < ;13
129	95, 96, 48, 126, 128	ndvdsi 12716	. 2 ⊢ ¬ ;13 ∥ ;;163
130	1, 48	decnncl 9804	. . 3 ⊢ ;17 ∈ ℕ
131		10nn 9800	. . 3 ⊢ ;10 ∈ ℕ
132		eqid 2238	. . . 4 ⊢ ;17 = ;17
133		eqid 2238	. . . 4 ⊢ ;10 = ;10
134	86	mullidi 8329	. . . . . 6 ⊢ (1 · 9) = 9
135		6p1e7 9445	. . . . . . 7 ⊢ (6 + 1) = 7
136	121, 27, 135	addcomli 8472	. . . . . 6 ⊢ (1 + 6) = 7
137	134, 136	oveq12i 6097	. . . . 5 ⊢ ((1 · 9) + (1 + 6)) = (9 + 7)
138		9p7e16 9877	. . . . 5 ⊢ (9 + 7) = ;16
139	137, 138	eqtri 2259	. . . 4 ⊢ ((1 · 9) + (1 + 6)) = ;16
140		9t7e63 9912	. . . . . . 7 ⊢ (9 · 7) = ;63
141	86, 120, 140	mulcomli 8333	. . . . . 6 ⊢ (7 · 9) = ;63
142	141	oveq1i 6095	. . . . 5 ⊢ ((7 · 9) + 0) = (;63 + 0)
143	2, 8	deccl 9795	. . . . . . 7 ⊢ ;63 ∈ ℕ0
144	143	nn0cni 9579	. . . . . 6 ⊢ ;63 ∈ ℂ
145	144	addridi 8469	. . . . 5 ⊢ (;63 + 0) = ;63
146	142, 145	eqtri 2259	. . . 4 ⊢ ((7 · 9) + 0) = ;63
147	1, 53, 1, 24, 132, 133, 69, 8, 2, 139, 146	decmac 9837	. . 3 ⊢ ((;17 · 9) + ;10) = ;;163
148		7pos 9408	. . . 4 ⊢ 0 < 7
149	1, 24, 48, 148	declt 9813	. . 3 ⊢ ;10 < ;17
150	130, 69, 131, 147, 149	ndvdsi 12716	. 2 ⊢ ¬ ;17 ∥ ;;163
151	1, 68	decnncl 9804	. . 3 ⊢ ;19 ∈ ℕ
152		eqid 2238	. . . 4 ⊢ ;19 = ;19
153		8cn 9392	. . . . . . 7 ⊢ 8 ∈ ℂ
154	153	mullidi 8329	. . . . . 6 ⊢ (1 · 8) = 8
155		7p1e8 9446	. . . . . . 7 ⊢ (7 + 1) = 8
156	120, 27, 155	addcomli 8472	. . . . . 6 ⊢ (1 + 7) = 8
157	154, 156	oveq12i 6097	. . . . 5 ⊢ ((1 · 8) + (1 + 7)) = (8 + 8)
158		8p8e16 9871	. . . . 5 ⊢ (8 + 8) = ;16
159	157, 158	eqtri 2259	. . . 4 ⊢ ((1 · 8) + (1 + 7)) = ;16
160		9t8e72 9913	. . . . 5 ⊢ (9 · 8) = ;72
161	53, 37, 38, 160	decsuc 9816	. . . 4 ⊢ ((9 · 8) + 1) = ;73
162	1, 69, 1, 1, 152, 80, 6, 8, 53, 159, 161	decmac 9837	. . 3 ⊢ ((;19 · 8) + ;11) = ;;163
163		1lt9 9513	. . . 4 ⊢ 1 < 9
164	1, 1, 68, 163	declt 9813	. . 3 ⊢ ;11 < ;19
165	151, 6, 66, 162, 164	ndvdsi 12716	. 2 ⊢ ¬ ;19 ∥ ;;163
166	37, 4	decnncl 9804	. . 3 ⊢ ;23 ∈ ℕ
167	120, 17, 56	mulcomli 8333	. . . . 5 ⊢ (2 · 7) = ;14
168	1, 7, 37, 167, 57	decaddi 9845	. . . 4 ⊢ ((2 · 7) + 2) = ;16
169	120, 32, 60	mulcomli 8333	. . . . 5 ⊢ (3 · 7) = ;21
170	37, 1, 37, 169, 61	decaddi 9845	. . . 4 ⊢ ((3 · 7) + 2) = ;23
171	37, 8, 37, 51, 53, 8, 37, 168, 170	decrmac 9843	. . 3 ⊢ ((;23 · 7) + 2) = ;;163
172		2lt10 9923	. . . 4 ⊢ 2 < ;10
173	50, 8, 37, 172	declti 9823	. . 3 ⊢ 2 < ;23
174	166, 53, 50, 171, 173	ndvdsi 12716	. 2 ⊢ ¬ ;23 ∥ ;;163
175	5, 12, 16, 20, 45, 47, 65, 94, 129, 150, 165, 174	prmlem2 13254	1 ⊢ ;;163 ∈ ℙ

This proof depends on syntax axioms:   ∈ wcel 2209  (class class class)co 6085  0cc0 8179  1c1 8180   + caddc 8182   · cmul 8184  2c2 9357  3c3 9358  4c4 9359  5c5 9360  6c6 9361  7c7 9362  8c8 9363  9c9 9364  ;cdc 9781  ℙcprime 12901

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-5 9368  df-6 9369  df-7 9370  df-8 9371  df-9 9372  df-n0 9568  df-z 9649  df-dec 9782  df-uz 9931  df-q 10029  df-rp 10065  df-fz 10422  df-fzo 10560  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-dvds 12571  df-prm 12902

This theorem is used by: (None)